With the desire to manage imprecise and vague information in evidential reasoning, several attempts have been made to generalize the Dempster-Shafer (D-S) theory to deal with fuzzy sets. However, the important principle of the D-S theory, that the belief and plausibility functions are treated as lower and upper probabilities, is no longer preserved in these generalizations. A generalization of the D-S theory in which this principle is maintained is described. It is shown that computing the degree of belief in a hypothesis in the D-S theory can be formulated as an optimization problem. The extended belief function is thus obtained by generalizing the objective function and the constraints of the optimization problem. To combine bodies of evidence that may contain vague information, Dempster's rule is extended by 1) combining generalized compatibility relations based on the possibility theory, and 2) normalizing combination results to account for partially conflicting evidence. Our generalization not only extends the application of the D-S theory but also illustrates a way that probability theory and fuzzy set theory can be integrated in a sound manner in order to deal with different kinds of uncertain information in intelligent systems.